Homework Help: Compare the speed and frequency of a wave in a thick rope

Why are no equations needed? I would need an equation or two to solve this question. Can you say what it/they might be? And then can you apply the equation(s) to check your answer?

 

The equation you posted is for a doubly-constrained rope, I believe, fixed at both ends. The problem in your OP is a little different. It seems to be asking about a long combination of two ropes, with the first part lighter (lower density) and the 2nd part heavier (higher linear density). The excitation at the left will generate transverse waves that will travel to the right with some frequency, velocity and wavelength. There will be a partial reflection of that traveling wave at the discontinuity between the two ropes, but you can ignore that for this question, I believe.

Now, what is the relationship between frequency, velocity and wavelength for a traveling wave? And if the left end of the light rope is being shaken up and down at some frequency, what would happen if the frequency changed at the place where the two ropes join?

After you think about that for a bit, check out this paper for information on a slightly different equation that you should consider...

http://www.cmp.caltech.edu/refael/league/wave-speed.pdf

And after reading that and thinking more about your answer, you can check out this nice summary post by @sophiecentaur in another thread about wave propagation and transitioning between different regions (in the case of that thread, it's EM waves and changing impedances of media that it is propagating through, but it still is very similar to this question of yours):

https://www.physicsforums.com/threads/refraction-of-a-wave.957400/#post-6071022